A key stimulus for the Bank of England’s review undertaken by Ben Bernanke (2024) was its serious under-estimation of UK inflation during 2021–2023. The Bank seemed to assume that the rise in inflation was transitory, possibly believing that inflation expectations were anchored around 2%, although similar serious forecast errors were made by other central banks. Bernanke’s 12 recommendations revealed worrying failings, on which Aikman and Barwell (2024) provide extensive commentaries.
A sequence of large one-sided one-step-ahead forecast errors as the forecast origin advances suggests a trend change. Forecasts can be ‘put back on track’ by impulse indicators acting as intercept corrections (ICs) at the forecast origin (Clements and Hendry 1996) as the value of the impulse indicator is the forecast error at that time point. Such forecast errors could be due to large outliers or measurement errors, step shifts in the mean of the process, or trend breaks. To isolate the source of a succession of large same-signed one-step-ahead forecast errors, and capture any sudden rapid shifts, in a recent paper (Castle et al. 2024) we use a deterministic-trend model and test if the first two or three large impulse indicators become insignificant when replaced by a broken linear or log-linear trend.
Using impulse indicators to correct forecast origin mis-forecasts has three advantages. First, they act as ICs, so the next forecast commences from the forecast origin with unchanged parameter estimates. This will lead to further large forecast errors if there is a location shift or trend shift, but a larger error after an outlier or measurement error compared to not adding an IC. Second, large forecast errors reveal that the current model needs updating, but too few new observations are available to do so for large systems. However, extending a deterministic-trend model by a broken trend is easily implemented. Finally, the adequacy of either a linear or log-linear trend extension can be tested by the resulting insignificance of the impulse indicators and an encompassing test against each other (see our paper for the underlying theory and Monte Carlo simulations). Because the new and old trends will increasingly diverge, ever-larger forecast errors will result if not corrected. Consequently, despite having few (only two or sometimes one or three) post-break observations, the new trend can be estimated reasonably accurately and continue to forecast adequately until another shift occurs.
The combination of the COVID-19 pandemic, supply chain disruptions and the energy crisis caused by Russia’s invasion of Ukraine led to several unanticipated rapid upswings in UK inflation. We show how quickly they could have been detected by modelling the log of monthly Consumer Prices Index including owner occupiers’ housing costs (CPIH) from the Office of National Statistics, denoted pt, with a data set covering 2010(1) to 2024(3). We follow a forecaster making one-step-ahead forecasts as the forecast origin advances each month from 2021(3) as the first of a multi-step sequence. The initial model up to 2021(3) explains pt by an intercept, linear trend, and trend indicator saturation (TIS; see Castle et al. 2019) selected at a significance level of 0.01%. Ten shifts between 2010(1) and 2021(3) were selected, correcting the overall trend to 0.071 (i.e. 0.85% per annum). Because all indicators are zero beyond their dates, the forecasts are simply the intercept and overall trend, which are 4.45 and 0.071, respectively, for the forecast for 2021(4) from 2021(3).
Figure 1(a) shows the fitted (dotted blue) and actual values (pt, solid red) and the resulting one-step-ahead forecast (with a ^, after the vertical line), and an interval forecast. Panel (b) shows the next 1-step forecast with an IC (˜, dotted) and without (^, dashed), and (c) records the next forecast with (^, dotted) and without (˜, dashed) a broken log-linear trend starting in 2021(3), estimated from the first two ICs with t-values of 3.2 and 5.4, which it eliminates. Finally, panel (d) extends the forecast horizon to 2021(9) with a multi-step root mean-square forecast error (RMSFE) of 0.15%, so despite the coefficient of the broken trend being estimated from just two observations, it forecasts better out of sample than the in-sample fit of 0.20%.
Figure 1 Successive forecasts of pt to 2021(9)
We use pt to test for and model changes in trend, but the forecasts for annual inflation (denoted ∆12pt in graph legends) can be derived from those by subtracting the log price level 12 months earlier. These will have the same error bars as the log levels, but re-centred on annual changes, shown in Figure 2 (a)–(d). That the first two forecasts in (a) and (b) are below the previous outcomes is all too common after shifts.
Figure 2 Successive forecasts of Δ12pt to 2021(9)
Re-estimating the model up to 2021(9) then forecasting 2021(10), a significant forecast error signals that a second break has happened, confirmed by another large same-sign forecast error for 2021(11) from 2021(10). Adding impulse indicators for 2021(9) and 2021(10) then forecasting 2021(12) from 2021(11) delivers a further large error, confirming it is not a step shift.
From Figure 1(d), the break probably started in 2021(8), so we create a broken linear trend starting then which makes the two impulse indicators insignificant, and eliminating them also removes the large forecast errors through to 2022(3), as can be seen in Figure 4 below.
However, forecasting 2022(4) from 2022(3) leads to another significant failure seen in Figure 3(a). By the time this shift is observed, Russia’s invasion of Ukraine and the consequent energy crisis and fuel and food price rises had occurred, so such a shift would not be a surprise, confirmed by another large error forecasting 2022(5) from 2022(4) (Panel (b)), offset by a broken log-linear trend starting in 2022(3) in Panel (c).
Figure 3 Successive forecasts of pt to 2023(9)
In fact, this model continues to forecast reasonably accurately through to 2023(9), which is 17 months ahead, as seen in Panel (d) confirming that medium-term forecasts can be usefully accurate despite estimating the most recent broken trend from just two observations, obviously conditional on no new breaks occurring.
We continue the multi-step forecast to 2023(9) in order to check if the approach could capture inflation first peaking then falling. The 17-steps-ahead forecasts for pt show that is indeed possible. Figure 4 plots all the sets of multi-step-ahead forecasts for both UK prices and annual inflation. Although the three broken trends in the model of pt estimated up to 2022(4) have positive coefficients, the annual inflation forecasts over 2022(5)–2023(9) capture the first eight falls after the downturn in inflation. This is partly an artefact of the previous year’s inflation being higher than the current one, but also requires accurate forecasts of pt. As no intermediate one-step-ahead forecasts yielded significant errors, we assume the forecaster continued with the same model.
When forecasting 2023(10) from 2023(9), a significant forecast error does occur revealing another trend break as the increases in pt slow. Once again, a broken log-linear trend from 2023(10) produces accurate forecasts over 2023(11)–2024(3), although fitted to just one non-zero observation. Figure 4 shows the multi-steps ahead forecasts of pt (Panel (a)) and ∆12pt (Panel (b)) over the inflation upsurge and fall from 2021(4)–2024(3), spanning four break episodes shown by vertical lines, with ellipses highlighting the breaks that had led to forecast failures. The Bank started raising interest rates from 0.1% in February 2022, continuing to raise in small steps till 5.25% in August 2023, yet our accurate 17-month-ahead (albeit ex-post) forecasts to 2023(9) were made in 2022(4), prior to most of those changes.
Had the method in Castle et al. (2024) been available before 2022(4), the same forecasts would have been made; one wonders what the MPC would have thought of them. Hendry and Muellbauer (2024) estimate the impacts on UK inflation in 2022 of import price inflation, energy shortages and price rises of 170%, suggesting they accounted for about 3/4 of the peak 9% rise in CPIH: as all these influences dropped considerably in 2023, UK inflation rose more slowly.
Figure 4 All four sets of successive forecasts of pt and Δ12pt to 2024(3)
The four-times-a-year projections for each December in the projections of the Monetary Policy Reports (MPRs) over the crisis period are shown in Figure 5 as symbols, and the earliest equivalent forecast from our approach by ∗ or #, usually made close to the second MPR projection. Dates are shown by vertical lines and the December annual inflation rate by horizontal lines, so forecast errors are measured by deviations of points from those horizontal lines. Our approach quickly detects rapid upsurges to avoid continuing forecast failure, rather than forecast from given dates, so is only relevant once a shift has led to large one-step forecast errors prompting the addition of broken trends.
Figure 5 Monetary Policy Report projections of annual inflation rates
The first such shift (shown as COVID, entailing a surge in demand at the end of lockdowns) was found in 2021(4), confirmed by 2021(5) enabling a log-linear trend to be estimated, and forecast till 2021(9) when a second shift (supply chain disruption) was detected in 2021(10). The MPR projection in February 2021 could not have known at that time of either shift, and even in May could not have known of the 2021(10) shift. Once the second shift was discovered, forecasts made for the end-year were accurate. The next shift in 2022(3) was again after the MPR’s first projection, but detecting it by 2022(5), our approach provides a much more accurate end-year forecast relative to the Monetary Policy Reports’ remaining dates (indeed, through till 2023(9)), avoiding the MPR erratic over-shoots. Conversely, forecasting the end 2023 inflation rate is less accurate as the shift in 2023(10) occurred late on.
Conclusion
A sudden unanticipated upsurge in a trending variable will usually create a sequence of large same-sign one-step-ahead forecast errors as the forecast origin advances. An application to the UK annual inflation upsurge since 2021 illustrates four trend breaks that were rapidly detected with forecasts greatly improved by adding either linear or log-linear broken trends. These four essentially unpredictable shifts are clearly visible in Figure 4, followed by significant forecast errors, but by rapidly detecting these, there were only seven large forecast errors overall in 36 forecasts rather than long periods of systematic forecast failure from the inflation upsurge and fall. Thus, the Bank could usefully add this approach to its suite of models to avoid future systematic forecast failure from unexpected shifts. ‘A trend is your friend till it doth bend’, but it has done so all too frequently over the last few years.
Such an approach could quickly detect rapid increases and ‘tipping points’ at the start of their evolution, acting both as an early-warning system and providing a glimpse of the road ahead, albeit without knowing when the next failure will occur.
References
Aikman, D and R Barwell (eds) (2024), The Bernanke Review: Responses from Bank of England Watchers, King’s College London.
Bernanke, B (2024), “Forecasting for monetary policy making and communication at the Bank of England: a review”, Bank of England Review, 12 April.
Castle, J L, J A Doornik, and D F Hendry (2024), “Forecasting after the start of a trend break”, Working paper, Nuffield College, Oxford University.
Castle, J L, J A Doornik, D F Hendry, and F Pretis (2019), “Trend-indicator saturation”, Working paper, Nuffield College, Oxford University.
Clements, M P and D F Hendry (1996), “Intercept corrections and structural change”, Journal of Applied Econometrics 11: 475–494.
Hendry, D F and J N J Muellbauer (2024), “Why did the Bank of England need a review of its forecasting record?”, Economic Observatory.
